Refined triangular element for laminated elastic–piezoelectric plates

www.elsevier.com/locate/compstruct

Composite Structures 78 (2007) 129–139

Refined triangular element for laminated elastic–piezoelectric plates q

Wu Zhen, Chen Wanji *

State Key Laboratory for Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116023, China

Available online 29 September 2005

Abstract

A refined three-noded triangular element based on the mixed theory is developed for the coupled-field analysis of laminated platescontaining piezoelectric layers, which satisfies the interelement C1 weak-continuity conditions. The concept of mixed theory is thatthe mechanical component is modeled by the global–local higher-order theory proposed by [Li XY, Liu D. Generalized laminate theoriesbased on double superposition hypothesis. Int J Numer Methods Eng 1997;40:1197–1212], which satisfies the free surface conditions andthe geometric and stress continuity conditions at interfaces whereas the electric field is modeled with layerwise theory. The numericalexamples of coupled and uncoupled analysis show that the transverse shear stresses can be accurately computed by the constitutive equa-tions and the present finite element method is suitable in predicting coupled behaviors of laminated plates under mechanical and electricloadings.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Mixed theory; Global–local higher-order shear deformation theory; Layer-wise theory; Refined triangular plate element; Interlaminar stresses;Coupled behavior; Mechanical–electric behavior

1. Introduction

Composite laminates with embedded or surface-bondedpiezoelectric layers are widely used in aeronautical andaerospace construction. So there has been a great deal ofinterest in study of these smart structures. Due to coupledeffects of mechanic and electric properties of piezoelectricmaterials, an accurate prediction of behavior of compositelaminates with surface-bonded piezoelectric layers willencounter a lot of challenges. Thus, the design and analysisof these complicated coupled systems require the develop-ment of new theories and finite element models.

Usually, the laminated plate theory is used to analyzethe problems of piezoelectric plates. For structural analysisof conventional or non-piezoelectric laminates, Pagano [1]firstly presented the exact solution of the laminates in cylin-drical bending. Based on this methodology, Ray et al. [2,3]

0263-8223/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2005.08.018

q Contract/grant sponsor: National Natural Sciences Foundation ofChina (No. 10172023).

* Corresponding author.E-mail address: [email protected] (C. Wanji).

and Heyliger et al. [4,5] present the exact solution of lami-nated plates composed of piezoelectric layers. Allik andHughes [6], Tzou and Tseng [7] and Xu et al. [8] have usedthree-dimensional finite element to solve the problem.However, the three-dimensional analysis requires hugecomputing, and it will encounter challenges when piezo-electric layers are too thin compared to the intermediatecore. In order to overcome these limitations, Garcaoet al. [9], Garcia Lage et al. [10,11] and Heyliger et al.[12,13] used layerwise theory instead of three-dimensionalfinite element models. Numerical results show that layer-wise theories give better accuracy but the number of un-known is dependent on the number of layers, which isalso rather expensive approach. To improve the computa-tional efficiency, Mitchell and Reddy [14] and Sheikh et al.[15] have presented the concept of mixed theory wheremechanical components are modeled with the single layerplate theory whereas electrical potential is modeled withlayerwise theory. The approach effectively achieves econ-omy by using single layer plate theory for mechanical com-ponents, for the number of unknowns is independent onthe number of layers. However, single layer plate theory

mailto:[email protected]

Fig. 1. Schematic figure for the laminated plate segment.

130 W. Zhen, C. Wanji / Composite Structures 78 (2007) 129–139

is not of the layer-wise type, which does not satisfy thecontinuity conditions of transverse shear stresses betweenlayers. An improvement in this approach is made by Choand his associates [16–18] and Topdar et al. [19] who haveused zig–zag theory instead of single layer plate theory tomodel the structural deformation. In Cho�s theory, thenumerical examples of coupled and uncoupled analysisshow that in-plane stresses compared with three-dimen-sional solutions are very accurate. However, the transverseshear stresses from the direct constitutive equations cannotbe calculated accurately. In order to accurately predictinterlaminar stress, the post-processing approach of inte-gration of equilibrium equation has to be adopted. Topar�smodel can predict accurately interlaminar stresses for thinand moderately thick laminates by directly constitutiverelations.

Based on the idea of mixed plate theory, an attempt hasbeen made in this paper to use the global–local higher-or-der shear deformation theory proposed by Li and Liu[20] to represent the structural deformation. The mainobjective of the present work is to study the capabilitiesof a refined element model based on the mixed plate theoryto capture the response details of piezoelectric plates. Theglobal–local higher order theory is composed of both glo-bal displacements and local displacements. By enforcingfree conditions of the transverse shear stresses on the upperand lower surfaces, and displacements and transverse shearstresses continuity conditions at the interfaces, the numberof unknowns can be reduced to 13. Based on the mixed the-ory, a refined triangular plate element is presented. This re-fined element satisfies the interelement C1 weak-continuityconditions. Moreover, this element only contains threenodes and each node has 13 unknowns for structural defor-mation whereas the number of unknowns for electric po-tential depends on the number of layers. It is greatlysignificant that transverse shear stresses can be calculatedaccurately by direct constitutive equation method. Never-theless, transverse normal stresses may be computed bythe equilibrium equation method. It should be indicatedthat the present theory is suitable for the problem of mod-erately thick smart laminates.

2. Mathematical formulation

2.1. The global–local higher-order shear deformation

theory for mechanical component

The global–local higher-order shear deformation theoryfor cross-ply laminated plates, which are called 1,2-3 dou-ble-superposition theory, has been presented by Li andLiu [20]. Based on this theory, the global–local higher-or-der theory for angle-ply laminated plates has been derivedby Wu et al. [21]. According to Li�s method, the local com-ponents can be divided into two groups: one group includesthe first- and second-order terms whereas another groupcontains the third-order term. The global–local higher-or-der theory may be simply derived as follows:

ukðx; y; zÞ ¼ uGðx; y; zÞ þ �ukLðx; y; zÞ þ uk

Lðx; y; zÞ ð1aÞvkðx; y; zÞ ¼ vGðx; y; zÞ þ �vk

Lðx; y; zÞ þ vkLðx; y; zÞ ð1bÞ

wkðx; y; zÞ ¼ wGðx; y; zÞ ð1cÞ

where uG, vG and wG are of global components; �ukL and �vk

L

are of two-term local groups; ukL and vk

L are of one-term lo-cal group; the superscript k represents the layer order oflaminated plates. The global components may be written as

uGðx; y; zÞ ¼ u0ðx; yÞ þ zu1ðx; yÞ þ z2u2ðx; yÞ þ z3u3ðx; yÞ ð2aÞvGðx; y; zÞ ¼ v0ðx; yÞ þ zv1ðx; yÞ þ z2v2ðx; yÞ þ z3v3ðx; yÞ ð2bÞwGðx; y; zÞ ¼ w0ðx; yÞ ð2cÞ

the local components can be written as

�ukLðx; y; zÞ ¼ fkuk

1ðx; yÞ þ f2kuk

2ðx; yÞ ð3aÞ�vk

Lðx; y; zÞ ¼ fkvk1ðx; yÞ þ f2

kvk2ðx; yÞ ð3bÞ

ukLðx; y; zÞ ¼ f3

kuk3ðx; yÞ ð3cÞ

vkLðx; y; zÞ ¼ f3

kvk3ðx; yÞ ð3dÞ

where

fk ¼ akz� bk; ak ¼2

zkþ1 � zk; bk ¼

zkþ1 þ zk

zkþ1 � zk

The relations between global coordinate and local coor-dinate can be seen in Fig. 1.

There are 6n + 9 degrees of freedom in the above dis-placements. By enforcing the displacement continuity con-ditions proposed by Li and Liu [20], 4(n � 1) degrees offreedom can be constrained. By imposing the continuityconditions of transverse shear stresses at the interface be-tween kth layer and (k � 1)th layer, 2(n � 1) degrees offreedom can be limited.

At present, the number of total degrees of freedom is re-duced to 15. The free conditions of the transverse shearstresses on the upper and lower surfaces are employed, fi-nally, and the number of the independent unknowns is re-duced from 15 to 11. Thereby, the ultimate displacementmodels for angle-ply laminated plates are

uk ¼ u0 þ Uk1ðzÞu1

1 þ Uk2ðzÞu1 þ Uk

3ðzÞu2 þ Uk4ðzÞu3 þ Uk

5w0;x

þ Uk6ðzÞv1

1 þ Uk7ðzÞv1 þ Uk

8ðzÞv2 þ Uk9ðzÞv3 þ Uk

10w0;y

ð4aÞ

W. Zhen, C. Wanji / Composite Structures 78 (2007) 129–139 131

vk ¼ v0 þ wk1ðzÞu1

1 þ wk2ðzÞu1 þ wk

3ðzÞu2 þ wk4ðzÞu3 þ wk

5w0;x

þ wk6ðzÞv1

1 þ wk7ðzÞv1 þ wk

8ðzÞv2 þ wk9ðzÞv3 þ wk

10w0;y

ð4bÞwk ¼ w0 ð4cÞwhere Uk

i and wki are the function of material constants and

thickness of laminated plate. The expression of Uki and wk

i isgiven in [21].

2.2. The layer-wise theory for electric potential

Sheikh et al. [15], Cho and his associates [16–18] andTopdar et al. [19] assumed that the electric potential ispiecewise linear along the transverse direction, and goodresults are obtained. Based on these successful experience,linear potential is used in this work.

By using the conventional piecewise linear assumptionof the electric potential across the thick direction, the elec-tric potential between z = zj and zj+1 can be written as

/j ¼ /jðx; y; zÞ ¼ Njjþ1/jþ1 þ N j

j/j ð5Þand thus

/ ¼ /ðx; y; zÞ ¼Xn

j¼1

wj/j ¼

Xn

j¼1

wjðNjjþ1/jþ1 þ N j

j/jÞ ð6Þ

where

Njj ¼

z� zj

zjþ1 � zj; Nj

jþ1 ¼zjþ1 � zzjþ1 � zj

wj ¼1 z 2 ½zj; zjþ1�0 otherwise

�

/j+1 = /j+1(x,y) and /j = /j(x,y) are the electric potentialat z = zj+1 and z = zj, respectively.

2.3. Constitutive relations

By incorporating the plane stress condition (rz = 0)through the thick direction, the constitutive equation ofpiezoelectric material in its material–axis system may be ex-pressed as [15]

rs ¼ Cses ð7Þwhere

rs ¼ r1 r2 r6 r4 r5 D1 D2 D3½ �es ¼ e1 e2 e6 e4 e5 E1 E2 E3½ �

Cs ¼Cm Cme

CTme Ce

� �

Cm ¼

c11 �c2

13

c33

c12 �c13c23

c33

0 0 0

c12 �c13c23

c33

c22 �c2

23

c33

0 0 0

0 0 c66 0 0

0 0 0 c44 0

0 0 0 0 c55

266666666664

377777777775

Cme ¼

0 0 e31 �c13e33

c33

0 0 e32 �c13e33

c33

0 0 0

e15 0 0

0 e24 0

266666666664

377777777775; Ce ¼

k11 0 0

0 k22 0

0 0 k33

2664

3775

In the above equations, ri and Di are the components ofstress and electric displacement respectively in the directionof i, ej and Ej are the components of strain and electric fieldrespectively in the direction of j and cij, kij and eij are thecorresponding elastic constant, permittivity and piezoelec-tric constant.

Using coordinate transformation, the constitutive equa-tion of a lamina in a common structural axis system may beexpressed as

r ¼ Qe ð8Þ

where

r ¼ rx ry sxy sxz syz Dx Dy Dz½ �T

e ¼ ex ey cxy cxz cyz Ex Ey Ez� �

Q ¼ T TCsT

In the above equation, T is the transformation matrix[15].

3. The transverse displacement function of element

The higher-order theory of the laminated plate possessessecond derivatives of transverse displacement w in thestrain components. The C1 continuity displacement func-tions should be used. Based on the refined non-conformingelement method proposed by Chen Wanji et al. [22,23], thedisplacement function w* satisfied C1 weak-continuity con-dition is adopted in this paper.

According to the refined element method [22], the dis-placement function w* satisfied C1 weak-continuity condi-tion may be expressed as

w� ¼ w0 þ PðBc � B0Þq ð9Þ

where P ¼ b 12x2 1

2y2 xy c

4. The refined three-node triangular plate element

4.1. The element transverse displacement function

satisfied C0 continuity condition

The famous nine-parameter non-conforming elementBCIZ [24] satisfies C0 continuity condition, which may bewritten as follows:

w0 ¼ Fq ð10Þ


where

F ¼ bF i; F xi; F yic; q ¼ fw0i ;w

0xi;w

0yig

T;

F i ¼ Li þ L2i Lj þ L2

i Lk � LiL2j � LiL2

k

F xi ¼ ckL2i Lj � cjL2

i Lk þ ðck � cjÞLiLjLk=2

F yi ¼ bjL2i Lk � bkL2

i Lj þ ðbj � bkÞLiLjLk=2

Li ¼ai þ bixþ ciy

2D; ai ¼ xjyk � xkyj

bi ¼ yj � yk; ci ¼ xk � xj; ði ¼ 1 � 3Þ

Second order derivatives of transverse displacement w0

may be given

o2w0

ox2

o2w0

oy2

o2w0

oxoy

� �T

¼ Bq ð11Þ

where B ¼ ½B1 B2 B3 �

Bi ¼

o2F i

ox2

o2F xi

ox2

o2F yi

ox2

o2F i

oy2

o2F xi

oy2

o2F yi

oy2

o2F i

oxoyo2F xi

oxoyo2F yi

oxoy

2666666664

3777777775ði ¼ 1 � 3Þ

B0 ¼1

D

Zve

Bdxdy ð12Þ

where D is element area.

4.2. The element transverse displacement function satisfied

C1 weak-continuity condition

In order to obtain the shape function of element trans-verse displacement satisfied C1 weak-continuity condition,the matrix Bc have to be given according to Eq. (9). Herethe matrix Bc can be given

Bc ¼ Bc1Bc2

Bc3½ � ð13Þ

Along the boundary 1–2 and boundary 3–1, o~won is ob-

tained by linear interpolation, o~wos is obtained by quadratic

interpolation. After integration, we have

Bc1 ¼

‘1m1 � ‘3m312ð‘2

1y21 þ ‘23y13Þ 1

2ð‘2

1x12 þ ‘23x31Þ

‘3m3 � ‘1m112ðm2

1y21 þ m23y13Þ 1

2ðm2

1x12 þ m23x31Þ

m21 � m2

312ð‘2

1x12 þ ‘23x31Þ 1

2ðm2

1y21 þ m23y13Þ

2664

3775

ð14Þwhere ‘i, mi are the cosines of the vector normal to the ithboundary. xij = xi � xj, yij = yi � yj, where xi and yi are thecoordinates of node i.

Bc2 and Bc3 can be obtained by the permutation of thesubscript. According to Eq. (9), the shape function F*

can be obtained

F � ¼ F þ P ðBc � B0Þ ð15Þ

The displacement functions of triangular element

u0 ¼X3

i¼1

Liu0i; u11 ¼

X3

i¼1

Liu11i

u1 ¼X3

i¼1

Liu1i; u2 ¼X3

i¼1

Liu2i

u3 ¼X3

i¼1

Liu3i; v0 ¼X3

i¼1

Liv0i

v11 ¼

X3

i¼1

Liv11i; v1 ¼

X3

i¼1

Liv1i

v2 ¼X3

i¼1

Liv2i; v3 ¼X3

i¼1

Liv3i;

w0 ¼X3

i¼1

ðF iw0;i þ F xiw0;xi þ F yiw0;yiÞ

w� ¼X3

i¼1

ðF �i w0;i þ F �xiw0;xi þ F �yiw0;yiÞ

/j ¼X3

i¼1

Li/ji ðj ¼ 1 � nþ 1Þ ð16Þ

where Li is area coordinate.

4.3. The strain matrix and stiffness matrix of three-node

triangular element

Based on linear strain–displacement relations, the strainfor the kth layer can be written as follows:

ek ¼ ouk ¼ B1 B2 B3½ �de ð17Þwhere de ¼ ½ de

1 de2 de

3 �T

dei ¼

u0i v0i w0i u11i u1i u2i u3i wx0i

v11i v1i v2i v3i wy0i /1 � � � /nþ1

� �i ¼ 1 � 3

o ¼

o

ox0

o

oyo

oz0 0 0 0

0o

oyo

ox0

o

oz0 0 0

0 0 0o

oxo

oy0 0 0

0 0 0 0 0o

oxo

oyo

oz

2666666666664

3777777777775

T

Here, strain matrix can be given directly

Bi ¼Bei 0

0 B/i

� �

where the matrix Bei is presented in detail in [21]:

B/i ¼

N 11

oLi

oxN 1

2

oLi

ox. . . N n

1

oLi

oxNn

2

oLi

ox

N 11

oLi

oyN 1

2

oLi

oy. . . N n

1

oLi

oyNn

2

oLi

oy

oN 11

ozLi

oN 12

ozLi . . .

oN n1

ozLi

oNn2

ozLi

26666664

37777775

Ni1, N i

2 are given in Eq. (6).

Fig. 2. An one-quarter square plate with a mesh of m · m.


The formation of strain matrix B of the refined non-con-forming element has been established. By using the follow-ing equation, the element stiffness matrix Ke may be given

Ke ¼Xn

i¼1

Z i

i�1

Z ZBTQiB dxdy

� dz ð18Þ

On the basis of the following equation, the vector of no-dal displacement de may be obtainedX

Kede ¼ r ð19Þ

where r is loading vector.After obtaining the vector of nodal displacement, the

nodal stresses can be obtained by utilizing direct constitu-tive equations.

5. Transverse normal stress from three-dimensional

equilibrium equation

Due to neglecting transverse normal strain in the presenttheory, it is impossible that the transverse normal stress iscalculated by the direct constitutive equation approach. Toobtain the normal stress, the equilibrium equation is used,and the formulation of transverse normal stress is given asfollows:

rz ¼ �Z z

�h2

osxz

oxþ osyz

oy

� dz ð20Þ

6. Numerical examples

In order to examine the adaptability and accuracy of thepresent refined element, several typical problems are ana-lyzed. Firstly, the problem on ordinary sandwich plate,which is used to validate capabilities of the refined elementto solve laminates with different ply thickness, is given inExample 1. Secondly, the problem of laminated plates sur-face bonded with piezoelectric layers, subjected to electricaland/or mechanical load, is analyzed in Examples 2 and 3.

In this section we select several typical analytical resultsfor comparison with the results of the present three-nodetriangular element PT. They are

Pagano [1]. The three-dimensional elasticity results aregiven by Pagano [1].Heyliger [12]. The three-dimensional elasticity results aregiven by Heyliger [12].Cho (EQUIL) [17]. The analytical results are given byCho et al. [17] from integrating equilibrium equations.Ray [2]. The three-dimensional elasticity results aregiven by Ray [2].

Table 1Boundary conditions for Examples 1–3

Boundary conditions x = constant

Simply-supported boundary u0 ¼ v0 ¼ w0 ¼ v11 ¼ v1 ¼ v2 ¼ v3 ¼

Symmetric axis u0 ¼ v0 ¼ u11 ¼ u1 ¼ u2 ¼ u3 ¼ w0;x ¼

Due to symmetry, only one-quarter of the laminatedplate has been studied. The mesh configuration of m · m

can be seen in Fig. 2.

Example 1. A symmetric square sandwich plate, simplysupported on all the four edges and subjected to bi-directional sinusoidal transverse loading q ¼ q0 sinðpx=LÞsinðpy=LÞ, has been considered in this example. Thethickness of each face sheet is one tenth of the totalthickness of the sandwich plate. The material properties ofthe face sheet and core material are presented.

Face sheets: E1 = 172.4 GPa, E2 = E3 = 6.89 GPa,G12 = G13 = 3.45 GPa, G23 = 1.378 GPa, v12 = v13 = v23 =0.25.

Core material: EX = EY = 0.276 GPa, EZ = 3.45 GPa,GXZ = GYZ = 0.414 GPa, GXY = 0.1104 GPa, vXZ = vYZ =vXY = 0.25.

The stresses are normalized as follows:

�sxz ¼ sxzðL=2; 0; zÞh=q0L; �syz ¼ syzð0; L=2; zÞh=q0L;

�rz ¼ rzð0; 0; zÞ=q0; �rx ¼ rxð0; 0; zÞh2=q0L2

where L is the length of laminated plate; h is the thicknessof lamination. Boundary conditions are given in Table 1.

For all of the examples, a m · m mesh density is used,which can be seen in all figures and tables. Table 2 containsnon-dimensionalized deflections and stresses for Example1. The results obtained from the present refined elementmethods are compared with the exact solution of Pagano[1]. It can be seen that there is good agreement betweenthe present results and 3D-elasticity solutions. In Figs. 3–5, the in-plane and transverse stresses obtained by constitu-tive relations are given. Here again, the present solutions ofstresses agree very well with 3D-elasticity solutions [1]. Thetransverse normal stresses along the thickness are given in

y = constant

w0;y ¼ 0 u0 ¼ v0 ¼ w0 ¼ u11 ¼ u1 ¼ u2 ¼ u3 ¼ w0;x ¼ 0

0 u0 ¼ v0 ¼ v11 ¼ v1 ¼ v2 ¼ v3 ¼ w0;y ¼ 0

Table 2Deflection and stress at the important point of a simply supported sandwich plate under sinusoidal load

L/h Grid �w (L/2,L/2,h/2) �rx (L/2,L/2,h) �ry (L/2,L/2,h) �sxy (0,0,h) �sxz (0,L/2,h/2) �syz (L/2,0,h/2)

2 4 · 4 22.444 2.8245 0.3811 0.2301 0.1718 0.12678 · 8 22.736 2.909 0.4003 0.2403 0.1822 0.136216 · 16 22.787 3.009 0.4141 0.2431 0.1853 0.1390Pagano [1] – 3.278 0.4517 0.2403 0.185 0.1402

4 4 · 4 7.5061 1.4481 0.2372 0.1382 0.2202 0.09428 · 8 7.6356 1.5004 0.2486 0.1460 0.2330 0.1030Pagano [1] 7.5968 1.556 0.259 0.148 0.239 0.107

10 4 · 4 2.1351 1.062 0.1035 0.0667 0.2782 0.03658 · 8 2.1895 1.1291 0.1084 0.0704 0.2928 0.0480Pagano [1] 2.2 1.153 0.110 0.072 0.3 0.053

20 4 · 4 1.1689 1.0119 0.0656 0.0476 0.2909 0.00788 · 8 1.2134 1.0898 0.0693 0.0504 0.3112 0.0275Pagano [1] – 1.110 0.07 0.051 0.317 0.036

Fig. 3. In-plane stress ð�rxÞ through the thickness of laminated plate forExample 1 (L/h = 4).

Fig. 4. Transverse shear stress ð�sxzÞ through the thickness for Example 1(L/h = 4).

Fig. 5. Transverse shear stress (�syzÞ through the thickness for Example 1(L/h = 4).

Fig. 6. Transverse normal stress ð�rzÞ through thickness for Example 1 (L/h = 4).


Fig. 6, which are calculated by three-dimensional localequilibrium equations.

Example 2. A smart laminated plate (p/90/0/90/p), simplysupported on all the four edges, is analyzed. The piezo-electric layers are PVDF, the thickness of which is one

tenth of the total thickness of the smart laminates and eachgraphite–epoxy layer is of equal thickness. Two load caseshave been studied by Heyliger and his associates [12]. Incase 1, a transverse load is applied at the top surface with apeak value of 1.0 N/m2. In this case, the top and bottomsurfaces of this plate are grounded. In case 2, a sinusoidalunit potential is applied to the top surface of the laminate,which is grounded only at the bottom surface. Moreover,


the four edges through the thickness are grounded in boththe cases. The material constants are given as follows:

Piezoelectric layers: C11 = 238 GPa, C22 = 23.6 GPa,C33 = 10.6 GPa, C12 = 3.98 GPa, C13 = 2.19 GPa, C23 =1.92 GPa, C66 = 6.43 GPa, C44 = 4.4 GPa, C55 = 2.15GPa, e31 = �0.13 C/m2, e32 = �0.14 C/m2, e33 = �0.28C/m2, e24 = e15 = �0.01 C/m2, k11/k0 = 12.5, k22/k0 =11.98, k33/k0 = 11.98, k0 = 8.85 · 10�12 C2/Nm2.

Graphite–epoxy layers: C11 = 134.9 GPa, C22 = 14.35GPa, C33 = 14.35 GPa, C12 = 5.156 GPa, C13 = 5.156GPa, C23 = 7.133 GPa, C66 = 5.654 GPa, C44 = 5.654GPa, C55 = 3.606 GPa, k11/k0 = 3.5, k22/k0 = 3, k33/k0 = 3, k0 = 8.85 · 10�12, C2/Nm2.

Table 3Central deflection w of a smart plate under mechanical load and electricpotential load

L/h Mesh Case 1 (w · 1011) Case 2 (w · 1012)

4 4 · 4 3.5259 0.19058 · 8 3.5975 0.195212 · 12 3.6086 0.1960Exact [12] 0.355274 0.220062

Fig. 7. In-plane displacement ð�uÞ through the thickness of laminated platefor Example 2 (L/h = 4).


The boundary conditions of displacement fields of thisexample are given in Table 1. However, the boundary con-ditions of electric fields may be written as follows: / = 0 atthe four edges across the entire thickness of smart lami-nated plate.

The displacements and stresses are normalized asfollows:

�u ¼ uðL=2; 0; zÞ � 1012; �rx ¼ rxð0; 0; zÞ;�sxy ¼ sxyðL=2; L=2; zÞ; �syz ¼ syzð0; L=2; zÞ;

Fig. 9. In-plane stress ð�sxyÞ through the thickness for Example 2 (L/h = 4).

Fig. 10. Transverse shear stress ð�syzÞ through the thickness for Example 2(L/h = 4).

Fig. 11. Electrostatic potential (/) through the thickness for Example 2.

Table 4Displacement and stresses for mechanical–electric coupling load comparison between the present and exact solution

h/L Grid V (volt) �u (z = h) (z = 0) �w (z = h/2) �rx (z = h) (z = 0) �ry (z = 2h/3) (z = h/3) �sxy (z = h) (z = 0) �sxz (z = h/2) �syz (z = h/2) �rz (z = h/2)

4 4 · 4 100 0.334 29.4331 �26.4 �20.502 2.606 �3.612 �0.721 �7.646�0.138 10.848 6.204 �0.703

8 · 8 0.347 30.481 �28.24 �21.56 2.70 �3.416 �0.699 �6.672�0.144 11.65 6.07 �0.737

Ray [2] 0.358 �31.56 �29.26 �21.24 2.706 �3.625 �0.91 �6.68�0.146 11.78 5.21 �0.728

Cho (EQUIL) [17] – �30.6 13.03 �25.1 �1.153 �3.606 �2.163 �8.363

10 4 · 4 100 0.0319 �2.091 �2.826 �2.121 0.175 �0.712 0.4341 �1.018�0.0213 1.333 �0.214 0.057

8 · 8 0.0339 2.191 �3.047 �2.29 0.182 �0.719 0.425 �0.949�0.0221 1.441 �0.237 0.0598

Ray [2] 0.0388 �2.35 �3.12 �2.34 0.181 �0.683 0.336 �0.878�0.0186 1.48 �0.261 �0.058

Cho (EQUIL) [17] – �2.22 1.49 �2.43 �0.072 �0.686 0.305 �0.932

100 4 · 4 100 �0.0057 0.3765 �0.456 0.148 �0.0176 0.2487 0.240 �2.540.00574 0.468 �0.171 0.0184

8 · 8 �0.0063 0.4053 �0.496 0.1574 �0.0196 0.409* 0.0502* 0.351*

0.0063 0.509 �0.183 0.0204Ray [2] �0.0063 0.447 �0.504 0.158 �0.019 0.382 0.086 0.486

0.0065 �0.518 �0.184 0.021Cho (EQUIL) [17] – 0.412 �0.518 0.158 0.021 0.382 0.086 0.485

* The 24 · 24 mesh density is adopted.

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Fig. 14. In-plane stress ð�ryÞ through the thickness of laminated plate forExample 3 (L/h = 4).


In Table 3, the present values of deflection are givenwith exact solutions [12]. The table shows that the presentresult obtained in the case of mechanical loading agreesvery well with the exact solutions. For the case of electricloading, however, the present result has a little error incomparison with those given by three-dimensional analysis.This is probably due to not considering the variation of w

along the thickness of the laminated plate in the presenthigher-order theory.

The present in-plane displacement and stresses formechanical loading are plotted with the exact results inFigs. 7–9. The transverse shear stresses obtained throughdirectly constitutive equation are compared with other re-sults in Fig. 10. Numerical results show the present resultsare good agreement with the exact results [12]. At the sametime, the electric potential across thickness at the plate cen-ter, for the electric loading, is given in Fig. 11. Very goodagreement is found between the present finite element solu-tions and the exact solutions.

Example 3. A simply supported square laminate withsurface-bonded piezoelectric layers, subjected to electricaland mechanical loading, is analyzed in this case. Thepotential of elastic core is made zero by grounding theelastic piezoelectric interfaces. The analytical solution ofthis problem has been given by Ray et al. [2]. A cross-plylaminated plate (0/90/0) consisting of three identicallaminate each of 3 mm thickness has been used as thesubstrate of the structure. The material constants of thesubstrate are as follows:

E1/E2 = 25, E3 = E2, G12 = G13 = 0.5E2, G23 = 0.2E2,v12 = v13 = v23 = 0.25, E2 = 6.9 GPa, k11 = k22 = k33 =8.85 · 10�12 F/m.

The thickness of each of the each piezoelectric layersbonded on the top and bottom surfaces of the laminate is0.04 mm and it has the following material properties:

EX = EY = EZ = 2 GPa, vXY = vXZ = vYZ = 0.29, k11 =k22 = k33 = 0.1062 · 10�9 F/m, e31 = e32 = 0.0046 C/m2,e33 = e24 = e15 = 0.0.

Fig. 12. Values of / to make the midplane deflection zero for various spanto thickness ratios.

The mechanical–electric load is doubly sinusoidal and itis applied at the top surface. The peak values of load at thecenter of the plate are q0 = 1 and V = 100.

The boundary conditions of displacement fields of thisexample are given in Table 1. However, the boundary con-ditions of electric fields may be written as follows: / = 0 atthe four edges across the entire thickness of smart lami-nated plate and the elastic core.

Fig. 15. In-plane stress (�sxyÞ through the thickness for Example 3 (L/h = 4).


The displacements and stresses are normalized

�w ¼ 100E2h3wð0; 0; zÞ=q0L4; �u ¼ E2h2uðL=2; 0; zÞ=L3;

ð�rx; �ryÞ ¼ ðrx; ryÞð0; 0; zÞh2=q0L2; �rz ¼ rzð0; 0; zÞ=q0;

�sxy ¼ sxyðL=2; L=2; zÞh2=q0L2;

�sxz ¼ sxzðL=2; 0; zÞh=q0L; �syz ¼ syzð0; L=2; zÞh=q0L;

Under mechanical–electric load, the transverse displace-ment and stresses at the important location are presented inTable 4 with 3D-elasticity results [2]. It can be observed

Fig. 17. Transverse shear stress ð�sxzÞ through the thickness for Example 3(L/h = 4).

Fig. 18. Transverse shear stress ð�syzÞ through the thickness for Example 3(L/h = 4).

Fig. 16. Transverse normal stress ð�rzÞ through thickness for Example 3(L/h = 4).

that, for thick plate problems, there is good agreement be-tween the present results and exact solutions.

Fig. 12 illustrates such a value of V, which makes thedeflection of the elastic core zero for different values oflength to thickness ratio. In Figs. 13–15, in-plane stressesacross the thickness for (L/h = 4) are given. Here again,the present results are in good agreement with 3D-elasticitysolutions. Transverse normal stresses are presented inFig. 16, which are obtained by integrating three-dimen-sional local equilibrium equations along the thickness oflaminate. It can be seen that present normal stresses agreevery well with exact solutions [2]. In Figs. 17 and 18, a littlevariation between the present transverse shear stresses cal-culated from constitutive relations and the exact solutionsis observed. This is probably that the thickness of piezoelec-tric layers is very small in comparison to the thickness of theelastic core, which is the only 0.0044% of that elastic core.

7. Conclusions

Based on the idea of mixed plate theory, an attempt hasbeen made in this paper to use the global–local higher-or-der shear deformation theory proposed by Li and Liu[20] to represent the structural deformation. The mainobjective of the present work is to study the capabilitiesof a refined element model based on the mixed plate theoryto capture the response details of piezoelectric plates. Theglobal–local higher-order theory satisfies interlaminarstresses continuity conditions and transverse shear freeconditions of upper and lower surfaces. Moreover, the to-tal number of degrees of freedom is independent of the lay-ers of laminated plate. At the same time, piecewise linearlayer-wise theory are adopted in the electric potential field.

Based on the mixed plate theory, the refined triangularelement has been developed. For this refined element, theinterelement C1 weak-continuity conditions can be satis-fied. Several typical numerical examples of decoupled/cou-pled responses have been analyzed and numerical resultsshow that the present refined element is very suitable foranalysis of interlaminar stresses of smart laminated plate.It is the most important that the in-plane stresses andtransverse shear stresses can be calculated accurately bythe constitutive relations. Furthermore, the transverse nor-mal stress may be also obtained accurately by integratingthe equilibrium equations.

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Refined triangular element for laminated elastic–piezoelectric plates